Download An Invitation to Geometric Quantization

An Invitation to Geometric Quantization
Alex Fok
Department of Mathematics, Cornell University
April 2012
What is quantization?
Quantization is a process of associating a classical mechanical system
to a Hilbert space. Through this process, classical observables are
sent to linear operators on the Hilbert space.
Example
Particle moving on R1 . The configuration space is the space of all
possible positions of the particle, which is R1 . The phase space is
T ∗ R = {(q, p)|q ∈ R1 , p ∈ Tq∗ R1 }
where q is the position and p is the momentum.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
2 / 29
What is quantization?
Example
Suppose the particle is subject to a potential energy which depends
on q(an example is the simple harmonic oscillator). Then
p2
+ V (q) = constant
2m
Turning the crank of quantization,
H = L2 (R1 )
q 7→ Mx = Multiplication by x
d
p 7→ −i~ dx
p2
2m
2
~
+ V (q) 7→ − 2m
∆ + MV (Schrödinger operator)
Alex Fok (Cornell University)
Geometric Quantization
April 2012
3 / 29
Symplectic manifolds
Definition
(X , ω) is a symplectic manifold if
X is a manifold
ω is a closed, non-degenerate 2-form
A compact symplectic manifold (X , ω) plays the rôle of a classical
phase space.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
4 / 29
Symplectic manifolds
Example
(X , ω) = (S 2 , area form).
ωp (ξ, η) = hξ × η, nbp i
Two questions:
Can we always quantize any (X , ω)?
What is the corresponding Hilbert space?
Alex Fok (Cornell University)
Geometric Quantization
April 2012
5 / 29
First attempt
We want the Hilbert space to be a certain space of sections of a
complex line bundle L on X equipped with a connection ∇ and a
covariant inner product h, i such that
curv(∇) = ω
So this imposes a condition on ω already.
Proposition
[ω] ∈ H 2 (X , Z) iff there exists (L, ∇, h, i) such that curv(∇) = ω.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
6 / 29
First attempt
Definition
(X , ω) is prequantizable if [ω] ∈ H 2 (X , Z).
Remark
[ω] = c1 (L). The class of ω is topological in nature and does not
depend on ∇.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
7 / 29
First attempt
Definition
(L, ∇, h, i) are prequantum data of (X , ω) if
curv(∇) = ω
h, i is covariant under ∇
The Hilbert space is
Z
n
ω
H = s ∈ Γ(L) hs, si
< +∞
n!
X
Alex Fok (Cornell University)
Geometric Quantization
April 2012
8 / 29
First attempt
Given a function f ∈ C ∞ (X ), what is the associated operator Qf ?
Definition
Xf is the symplectic vector field such that
ιXf ω = df
Definition
Qf = ∇Xf + if
Alex Fok (Cornell University)
Geometric Quantization
April 2012
9 / 29
First attempt
Proposition
Qf is skew-adjoint with respect to the inner product hh, ii on Γ(L)
defined by
Z
ωn
0
hhs, s ii = hs, s 0 i
n!
X
Alex Fok (Cornell University)
Geometric Quantization
April 2012
10 / 29
A classical example
Example
Let X = S 2 the unit sphere in R3 centered at the origin,
ω = Area form. If
L = TS 2
∇ = Riemannian connection induced from that of T R3
h, i = Riemannian metric
Then (L, ∇, h, i) are prequantum data.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
11 / 29
A classical example
Example
We have
Xf = Tangent vectors of latitudes
More precisely, if p = (ϕ, θ) in spherical coordinates,
(Xf )p = sin ϕ(cos θ~i + sin θ~j)
Qf Xf = 0
Alex Fok (Cornell University)
Geometric Quantization
April 2012
12 / 29
Disadvantage of first attempt
H obtained from this quantization scheme is too large to handle.
One way to get around this is to introduce polarization and
holomorphic sections to cut down the dimensions of H. Then we
need to impose more structures on the compact symplectic manifold.
A natural candidate: Kähler manifold.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
13 / 29
Kähler manifolds
Definition
(X , ω, J) is a Kähler manifold if
ω is a symplectic 2-form.
J is an integrable almost complex structure, i.e. X is a complex
manifold and J corresponds to multiplication by i on each fiber
of T U where U is a holomorphic chart.
ω and J are compatible in the sense that ω(·, J·) is positive
definite.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
14 / 29
Examples of Kähler manifolds
Example
√
√
X = Cn = {(x1 + −1y1 , · · · , xn + −1yn )|xi , yi ∈ R, 1 ≤ i ≤ n}.
n
n
Identifying
√ Tp C with C , letting ei be the i-th standard basis vector
and fi = −1ei , we define J by
Jei = fi , Jfi = −ei
and
ω=
n
X
dxi ∧ dyi
i=1
n
Then (C , ω, J) is a Kähler manifold. Actually any Kähler manifold
locally looks like (Cn , ω, J).
Alex Fok (Cornell University)
Geometric Quantization
April 2012
15 / 29
Examples of Kähler manifolds
Example
π
2
counterclockwise on tangent spaces when viewing S 2 from outside, is
Kähler.
(S 2 , ω, J), where ω = area form and J is the rotation by
Example
CPn and any smooth projective subvarieties of CPn are Kähler.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
16 / 29
Examples of Kähler manifolds
Example
The coadjoint action of a compact Lie group G on g∗ is defined by
hAd∗g γ, ξi = hγ, Adg −1 ξi
Let Oγ be the orbit of the coadjoint action of G passing through
γ ∈ Int(Λ∗+ ). Then
Oγ ∼
= G /T , T being a maximal torus of G .
Tβ O γ ∼
= g/t.
Using the above identifications, we define a 2-form
ωβ (ξ, η) = β([ξ, η])
ω, called the Kostant-Kirillov-Souriau form, is symplectic and
integral.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
17 / 29
Examples of Kähler manifolds
Example
Consider the complexified Lie algebra gC and its root space
decmposition
M
gC = t C ⊕
gα
α∈R
Let {Hα , Xα }α∈R be the Chevalley basis of gC which satisfies
2hα, βi
[Hα , Xβ ] =
Xβ
hβ, βi
[Xα , X−α ] = Hα for α ∈ R +
Then
M
g/t =
spanR {eα , fα }
α∈R +
√
where eα = −1(Xα + X−α ), fα = Xα − X−α .
Alex Fok (Cornell University)
Geometric Quantization
April 2012
18 / 29
Examples of Kähler manifolds
Example
Define J by
Jeα = fα , Jfα = −eα
Then (Oγ , ω, J) is Kähler.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
19 / 29
Kähler polarization
Suppose dimC X = n. Consider the complexified tangent bundle
TX ⊗R C
Definition
A complex rank n subbundle F ⊆ TX ⊗R C is a positive-definite
polarization if
It is integrable, i.e. closed under Lie bracket.
For all X , Y ∈ F , ωC (X , Y ) = 0
√
−1ωC (·, ·) is positive-definite.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
20 / 29
Kähler polarization
Example
X = Cn . Then
Tp X ⊗R C ∼
= spanC {e1 , f1 , · · · , en , fn }
Then
F = spanC {e1 +
√
−1f1 , · · · , en +
√
−1fn }
is a positive-definite polarization.
F should be thought of as the ‘holomorphic direction’ in TX ⊗R C.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
21 / 29
Second attempt: Kähler quantization
Theorem
Let (X , ω, J) be a compact Kähler manifold with positive-definite
polarization F , and (L, ∇, h, i) be prequantum data. Let
Xquantum = {s ∈ Γ(L)|∇Θ s = 0 for all Θ ∈ F }
Then Xquantum is finite-dimensional.
We define the quantization of (X , ω, J) to be Xquantum , which is the
space of holomorphic sections of the prequantum line bundle L.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
22 / 29
Second attempt: Kähler quantization
Example
X = (Oγ , ω, J) as in a previous example. A positive-definite
polarization is
√
F = spanC {eα + −1fα }α∈R + = spanC {Xα }α∈R +
So F = spanC {X−α }α∈R + . Note that L = G ×T Cγ is a prequantum
line bundle. By Borel-Weil Theorem,
Xquantum = space of holomorphic sections of L
= Irreducible representation of G with highest weight γ
Alex Fok (Cornell University)
Geometric Quantization
April 2012
23 / 29
Second attempt: Kähler quantization
Example
X = (S 2 , ω, J) with L = TS 2 . Then Xquantum is a 3-dimensional
complex vector space. Identifying S 2 with C ∪ ∞ through
stereographic projection, we can describe three holomorphic vector
fields of S 2 which form a basis of Xquantum as follows
the vector field s0 generated by the infinitesimal action of
1 ∈ Lie(S 1 ) of the S 1 -action on C ∪ ∞ given by rotation
z 7→ e iθ z,
the vector field s−2 generated by the infinitesimal action of
1 ∈ Lie(R1 ) of the R1 -action on C ∪ ∞ given by translation
z 7→ z + a,
the vector field s2 generated by the infinitesimal action of
1
1 ∈ Lie(R1 ) of the R1 -action on C ∪ ∞ given by z 7→
z +a
Alex Fok (Cornell University)
Geometric Quantization
April 2012
24 / 29
Quantization of G -Kähler manifolds
One may further consider a compact Kähler manifold with
prequantum data and a nice G -action where G is a compact Lie
group. By nice we mean
G preserves both ω and J.
There exists a map called moment map
µ : X → g∗
which is G -equivariant(here G acts on g∗ by coadjoint action)
and
ιξ] ω = dhµ, ξi for all ξ ∈ g
We say G acts on X in a Hamiltonian fashion.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
25 / 29
Quantization of G -Kähler manifolds
Let
Qξ := Qhµ,ξi = ∇ξ] + ihµ, ξi
This gives a g-action on Γ(L). In nice cases, e.g. G is
simply-connected, this action can be integrated to a G -action. It
turns out that G acts on Xquantum , which makes it a G -representation.
Question: What can we say about the multiplicities of weights of this
representation?
Alex Fok (Cornell University)
Geometric Quantization
April 2012
26 / 29
Quantization commutes with reduction
Definition
Let G act on (X , ω, J) in a Hamiltonian fashion with moment map µ.
Assume that 0 is a regular value of µ and G acts on µ−1 (0) freely.
The symplectic reduction of X by G is defined to be
XG := µ−1 (0)/G
XG can be thought of as the fixed ‘points’ of the phase space X .
One can construct prequantum data (LG , ∇G , h, iG ) and
positive-definite polarization of XG from those of X by restriction and
quotienting.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
27 / 29
Quantization commutes with reduction
Theorem (Guillemin-Sternberg, ’82)
G
dim(Xquantum
) = dim((XG )quantum )
By virtue of this theorem, one can compute the multiplicity of the
trivial representation in Xquantum by looking at the quantization of XG .
Alex Fok (Cornell University)
Geometric Quantization
April 2012
28 / 29
Quantization commutes with reduction
Example
X = (S 2 , ω, J), G = S 1 acts on X by rotation around the z-axis,
with µ being the height function. Note that
e iθ · s0 = s0 , e iθ · s−2 = e −2iθ s−2 , e iθ · s2 = e 2iθ s2
So as S 1 -representations,
Xquantum ∼
= C0 ⊕ C−2 ⊕ C2
1
S
It follows that dim(Xquantum
) = 1. On the other hand,
XS 1 = a point
A line bundle over a point is simply a 1-dimensional vector space. So
dim((XS 1 )quantum ) = 1.
Alex Fok (Cornell University)
Geometric Quantization
April 2012
29 / 29